论文标题
通过无限基质产物近似板板的组代数
Approximating the group algebra of the lamplighter by infinite matrix products
论文作者
论文摘要
在本文中，我们在研究$*$ - 定期关闭某些特定组代数$ kg $内部的$ \ Mathcal {u}（g）$，$*$ - 无界操作员的代数$*$ $ kg $的研究中介绍了一种新技术。我们在这项研究中使用的主要工具是一类形式的交叉产品代数的一般近似结果在任意字段$ k $上。这类代数与合适的组代数之间的连接由傅立叶变换提供。利用这种机械，我们研究了灯泡组代数的明确近似。作者在另一篇论文中使用了这一点，以获取由Lamplighter组引起的$ \ ell^2 $ -BETTI数字的整个家庭，其中大多数是先验。
In this paper, we introduce a new technique in the study of the $*$-regular closure of some specific group algebras $KG$ inside $\mathcal{U}(G)$, the $*$-algebra of unbounded operators affiliated to the group von Neumann algebra $\mathcal{N}(G)$. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form $C_K(X) \rtimes_T \mathbb{Z}$, where $X$ is a totally disconnected compact metrizable space, $T$ is a homeomorphism of $X$, and $C_K(X)$ stands for the algebra of locally constant functions on $X$ with values on an arbitrary field $K$. The connection between this class of algebras and a suitable class of group algebras is provided by Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of $\ell^2$-Betti numbers arising from the lamplighter group, most of them transcendental.
